Based on the wording of the requirements for the Feynman Grand Prize, I assumed that three degrees of freedom (translational motions only) were all that was required - no rotations seem to be mentioned, so 6 degrees of freedom doesn't seem correct. What am I missing?

It was twenty years ago that I participated in the discussions on this, so my memory may be a bit cloudy, but I seem to remember that the original intent was to require rotations as well. It does say "...controlled motions needed to manipulate and assemble individual atoms or molecules into larger structures, with atomic precision", and as Drexler elaborates in Nanosystems, and subsequent simulations from Merkle and others show, that often requires rotations. IMO, it should say 6 DOF explicitly, but you are correct that it does not.

I do see that I misremembered the full adder. It is supposed to add two 8-bit numbers, not 4-bit numbers. So, that would take 15 half-adders, not 7. I don't know why it says to discard the overflow. It seems trivial to include it, and that makes the module much more useful. But, the task is simply to demonstrate construction capability, not to build an actually useful device, so that is of minor consequence.